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Self-dual code over \(R\)

Description

An additive linear code \(C\) over a ring \(R\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product.

For \(m=2^{s} p_{1}^{n_{1}} \cdots p_{r}^{n_{r}}\) with distinct odd primes \(p_i\), the group ring \(\mathbb{Z}_m G\) contains a self-dual group code if and only if all exponents \(n_i\) are even and either \(s\) or \(|G|\) is even [1; Thm. 16.12.6].

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(R\)-linear codeECC
Dual linear code over \(R\)
Parents
Dual linear code over \(R\)
Self-dual code over \(R\)
Children
Self-dual linear code
Self-dual linear codes are over fields, which are also rings.
Self-dual code over \(\mathbb{Z}_q\)

References

[1]
W. Willems, “Codes in Group Algebras.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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Zoo Code ID: self_dual_over_rings

Cite as:
“Self-dual code over \(R\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_over_rings
BibTeX:
@incollection{eczoo_self_dual_over_rings, title={Self-dual code over \(R\)}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/self_dual_over_rings} }
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Permanent link:
https://errorcorrectionzoo.org/c/self_dual_over_rings

Cite as:

“Self-dual code over \(R\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_over_rings

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/dual/self_dual_over_rings.yml.